suppose i want to model a 3D plane intersections when they make a corner. that is three or more than three planes will be intersected at some corners of cubes or other volumetric objects. i can find this corner point by considering as a plane intersection problem. also, i can do this by considering as line intersections problem. here, lines can be found by intrersecting two planes at a time.so, i would like to know this plane intersection and line intersection is given almost same result or two different results. suppose, i fitted planes by least square and found the best planes from the given set of points. now, i am confused and think these two method would be given 2 results. any comment please.
Here is what I have done in a similar situation.
Call your three planes $A$, $B$, $C$, and their lines of intersection
$ab$, $bc$, $ca$. Each pair of lines should intersect in a point, but likely they
slightly miss. A good candidate is the midpoint of the shortest segment connecting the
pair of lines. So, you could write a routine to compute that shortest segment (you can find the computation in many locations on the web, e.g., this StackOverflow question); let's call it $m(L_1,L_2)$ for two
lines $L_1$ and $L_2$. Then average the three points: